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Extra resources for A categori
cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors

Example text

Xn ∈ Z and ρn is the half-sum of the positive roots of gln . This direct sum is a gln−2 -module in a natural way. Define νn (M ) as the tensor product of this module with the one-dimensional gln−2 -module of weight e1 + · · · + en−2 . 232 J. Bernstein, I. Frenkel and M. Khovanov Sel. , New ser. Proposition 17. Functors ςn and νn are mutually inverse equivalences of cate1 gories Ok−1,n−k−1 and Ok,n−k . We omit the proof as it is quite standard. i , 1 ≤ i ≤ n−1 and Ok−1,n−k−1 are equivalent.

Let T i , Ti be translation functors on and off the i-th wall T i :Oµ −→ Oµi Ti : Oµi −→ Oµ . These functors are defined up to an isomorphism by the condition that they are projective functors between Oµ and Oµi and 1. Functor T i takes the Verma module Mµ to the Verma module Mµi . 2. Functor Ti takes the Verma module Mµi to the projective module Psi µ where si is the transposition (i, i + 1). On the Grothendieck group level, [Ti Mµi ] = [Mµ ] + [Msi µ ]. Let pk be the maximal parabolic subalgebra of gln such that pk ⊃ n+ ⊕ h and the reductive subalgebra of pk is glk ⊕ gln−k .